# Half Angle Calculator

> Calculate half angle trigonometric values using half angle formulas. Find sin(θ/2), cos(θ/2), and tan(θ/2).

**Category:** Math
**Keywords:** half angle, trigonometry, sin, cos, tan, angle, formula, identities, theta
**URL:** https://complete.tools/half-angle-calculator

## How it calculates

This calculator uses three key half-angle formulas: 

1. sin(θ/2) = ±√((1 - cos(θ)) ÷ 2)  
2. cos(θ/2) = ±√((1 + cos(θ)) ÷ 2)  
3. tan(θ/2) = sin(θ) ÷ (1 + cos(θ))  

Here, θ is your original angle. The ± sign changes depending on the quadrant where θ/2 falls. For instance, if θ is in the first quadrant, both sine and cosine values will be positive. These formulas allow you to compute the trigonometric functions of half the angle based on the cosine of the original angle, which simplifies complex calculations and helps derive trigonometric identities.

## Who should use this

This tool is perfect for mathematicians simplifying trigonometric expressions in calculus, architects who rely on trigonometry for structural designs, physicists studying wave interference patterns, and computer graphics developers needing to manipulate angles for rendering techniques.

## Worked examples

Let’s look at a couple of examples to see how this works:

Example 1: Want to find sin(15°)? We know sin(30°) = 0.5, so we can use the half-angle formula:

sin(15°) = √((1 - cos(30°)) ÷ 2)  
= √((1 - √3/2) ÷ 2)  
= √((2/2 - √3/2) ÷ 2)  
= √((2 - √3) ÷ 4)  
= (√(2 - √3)) ÷ 2

Example 2: Let’s calculate cos(45°). Here, θ = 90°:

cos(45°) = √((1 + cos(90°)) ÷ 2)  
= √((1 + 0) ÷ 2)  
= √(1/2)  
= √2 ÷ 2

Example 3: Now for tan(30°). Since tan(60°) = √3, we use:

tan(30°) = sin(60°) ÷ (1 + cos(60°))  
= (√3/2) ÷ (1 + 1/2)  
= (√3/2) ÷ (3/2)  
= √3 ÷ 3.

## Limitations

Keep in mind a few limitations of the Half Angle Calculator. First, it expects the input angle to be in degrees or radians—entering something else might cause errors. Also, there are precision limits when calculating square roots and trigonometric values, particularly for angles near critical points like 0° or 90°. The tool might not give accurate results for angles outside the standard range (0° to 360°) unless you convert them properly. Lastly, remember that the ± sign in the formulas depends on where the angle is located on the unit circle, so be cautious about that.

## FAQs

Here are some common questions:


**Q:** How does the sign of the half-angle formulas affect the result?  


**A:** The sign depends on the quadrant where the half-angle lies. In the first and fourth quadrants, values are positive; in the second and third quadrants, they're negative.


**Q:** Can the Half Angle Calculator handle angles greater than 360°?  


**A:** Yes, but it assumes angles are in standard position. You'll need to reduce any angles greater than 360° to fit within the standard range.


**Q:** Why are half-angle formulas useful in calculus?  


**A:** They make it easier to evaluate integrals and derivatives involving trigonometric functions, simplifying complex calculations.


**Q:** Are the results from the calculator exact or approximate?  


**A:** Results may vary slightly due to numerical precision based on the calculator's algorithms, especially for certain angles.

---

*Generated from [complete.tools/half-angle-calculator](https://complete.tools/half-angle-calculator)*